Math 4400/Number Theory/Fall 2012
Stuff to Know for the Third Midterm
Definitions you need to know.
(1) What is the Legendre symbol?
(2) What is Euler’s criterion for computing the Legendre symbol?
(3) What is a Gaussian integer?
(4) What is the norm of a Gaussian integer?
(5) What is an irreducible (indecomposable) Gaussian integer?
(6) What is Pell’s Equation?
How to’s.
(1) Find a field with p2 elements.
(2) Compute any Legendre symbol using Quadratic Reciprocity.
(3) Carry out Fermat’s infinite descent for sums of squares.
(4) Find the gcd of a pair of Gaussian integers.
(5) Factor a Gaussian integer.
(6) Given a solution to Pell’s equation, find infinitely many others.
Theorems you should be able to state precisely.
(1) Quadratic Reciprocity
(2) Unique factorization of Gaussian integers.
(3) Exactly which natural numbers are sums of two squares?
Stuff you should be able to prove.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Euler’s criterion.
−1 is a square mod p if and only if p = 2 or p ≡ 1 (mod 4).
The multiplicativity of the Legendre symbol.
2 is a square mod p if and only if p ≡ 1 or 7 (mod 8).
Any prime ≡ 1 (mod 4) is a sum of two squares.
Any prime ≡ 3 (mod 4) is not a sum of two squares.
The unit Gaussian integers are 1, −1, i, −i and nothing else.
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